The following problem is from Bird Transport phenomena.
I have solved this problem and I know the solution. But one part got me interested, which I cannot prove the statement and that is the first sentence of part c.
Can you help about the proof of non-existence of analytical solution for the differential equation 3.K-1?
Splution of $$f''+af^2=b$$ multiply by $2f'$ both sides to get $$2f'f''(x)=-2af'f^2+2f'b \implies (f'^2)'=(-2f^2+2b) f'.$$ Integrate both sides w.r.t. $x$, to get $$\int\frac{df'^2}{dx} dx=\int(-2af^2+2b)\frac{df}{dx} dx \implies f'^2=-\frac{2a}{3}f^3+2bf+A$$ $$\implies f'= \pm \sqrt{-2af^3+2bf+A} \implies \int \frac{df}{\sqrt{-2af^3+2bf+A}}=\pm\int dx+B.$$ The integral on the left relates to Elliptic integrals.